Lucky strikes

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As this is being written Comet Shoemaker-Levy 9 is but a few days away from striking Jupiter and offering us a bird’s-eye preview of what would happen if a comet collided with Earth. Not that we will see the actual impacts, for these occur just over the horizon on the night side of the planet, and so are hidden from us; however, the rotation of the planet will carry the site into view within half an hour of impact.

Periodicals of every sort have carried articles about this event, conjectures as to the results of the impacts, timetables for the several collisions and, of course. references to the great comet which American geologist Walter Alvarez used to mark the end of the Cretaceous period and the terminal decline of the dinosaurs. Even the astrological significance has been discussed, for if the planets influence us then they can hardly be knocked about without some, probably dire, consequence. (Though just why or how the arrival of a few tonnes of assorted ice and rock on Jupiter should have any perceptible effect on human affairs is totally obscure.)

If, in truth, such astrological beliefs have shaky foundations, then they are in distinguished company, for astronomers from the time of Newton to 1900, and a host of their amateur brethren since, have believed in determi­nable planetary orbits.

The first reformation of the classical picture of a geocentric cosmos was the reintroduction of a heliocentric system by Nicholas Copernicus. (Origi­nally. Aristarchus of Samos had unsuccessfully argued the case in around 265 e.c.) In De Revolutionibus, 1543, Copernicus again placed the Sun at the centre of the solar system, and this resulted in a greatly simplified set of circular mo­tions, although with no signifi­cant improvement in predictive accuracy. Fiddle with the system as they would, his successors  could not get complete agreement between their predictions and the observed planetary posi­tions. It took the vast corpus of Tycho Brahe’s remarkably accurate observations coupled with the persevering genius of Johannes Kepler to solve the problem.

After five years of unremitting toil, Kepler was able to demonstrate that Mars orbits the Sun in an ellipse rather than a circle. Thus he destroyed the 2000-year-old assump­tion that circles are funda­mental to the structure of the heavens. His three laws of planetary motion were a towering achievement, and provide the foundation of non-relativistic celestial mechanics to this day. For completeness, all they lacked was the mechanism which drove the machin­ery—the “why” as well as the “how”—and this was to be supplied by Newton’s Laws of Motion and Uni­versal Law of Gravitation.

To Kepler and his 17th century contemporaries, the contents of the cosmos had hardly changed since the days of Aristotle. Tycho’s observations and calculations had shown that comets move in interplanetary space rather than in the classical sublunary sphere where they could be explained away as atmospheric exhalations. Galileo’s telescope had revealed the four largest moons of Jupiter and the multitude of stars in the Milky Way. These observations in no way affected confidence in the Kepler/Newton achievement—apparently accurate prediction and complete understanding had been achieved.

At this point the devel­opment of astronomy followed two distinct paths. First, astronomers discovered there was a greater number and variety of objects in the solar system than had been previously thought, and second, the predictive model came under math­ematical attack.

In 1655 Huygens discovered the first of the five brightest satellites of Saturn, Titan, and in the next 30 years Cassini added four more. Then in 1781 William Herschel discovered a sixth planet, Uranus, and thereafter two of its moons, as well as two more around Saturn.

At the beginning of the nineteenth century Piazzi discovered a new class of object, the first of the asteroids—objects orbiting the Sun but too small to be thought of as planets in any accepted sense. Astronomers wondered whether they were associ­ated with meteors—shooting stars—which Chladni had shown as early as 1794 to be extra terrestrial.

Then in 1834 Biot verified the existence of meteorites—those meteors which have survived passage through the Earth’s atmosphere to reach the ground. Meteors and meteorites were then linked with the periodic comets by Le Verrier and Schiaparelli, who demon­strated the association of a meteor shower in 1866 with the track of Comet Temple (which had meanwhile vanished, presumably fragmented during its passage around the Sun.)

Thus the seven objects of the classical solar system (Sun, Moon and five planets) multiplied at an ever-increasing pace. Neptune and Pluto were added, so that there are now nine planets, and the Moon is but one of a total of 44 planetary satellites. The known asteroids are numbered by the thousand  and must total hundreds of thousands, the vast majority of them too faint to be seen with presently available techniques. The original “asteroid belt” lying between Mars and Jupiter has been aug­mented by three families of asteroids, the Apollos, the Amors and the Atens, orbiting the Sun with paths that cross that of the Earth. There are also groups orbiting beyond Saturn.

Comets, once thought to be occasional visitors from the depths of space, are now known to be only the visible part of a vast reservoir of icy fragments orbiting the Sun beyond Neptune. The inner part of this yet-to-be-observed congregation is the Kuiper Belt, lying roughly in the plane of the ecliptic between 40 and perhaps 1000 A.U. from the Sun. (1 Astronomical Unit is the mean distance of the Earth from the Sun, about 150,000,000 km.) Beyond this belt is the Oort Cloud, completely surrounding the solar system and ex­tending out 100,000 A.U., almost half way to Alpha Centauri, the nearest star.

Finally, the rings of Saturn have, thanks to the Voyager fly-by, been shown to be orbital systems of extraordinary complexity—and not unique; both Jupiter and Uranus have rings. Thus the majestic simplicity of Newton’s clockwork solar system has become some­thing far more complex, involving a huge number of gravitational interac­tions. To these we now add the effects of the planetary magnetospheres and the solar wind.

Now Newton had given strict mathematical form to Kepler’s Laws and de­duced precisely (having invented calculus) how any given body will orbit its primary given the initial conditions, the masses of the bodies, their separation and the instan­taneous velocity of one relative to the other. This calculation, called the Two Body Problem, applies to any two bodies orbiting one another and free of any other influence, and is, in the ideal world of math­ematics, exactly soluble. However, such a solution involves Newton’s Law of Universal Gravitation, which states that there exists a force of attraction between any two bodies which is proportional to the product of their masses and inversely proportional to the square of the dis­tance between them.

Here is the trap: “any two bodies” means exactly that, and not just the two bodies that you are inter­ested in. Not only must the force between A and B be considered but also that between A and C, D, E, F, etc, throughout the uni­verse, and likewise for B. Of course, Newton was aware of this, but because of the huge size of the Sun compared with the other bodies in the solar system he was able to treat each body relative to it alone with a quite acceptable degree of accuracy for his purposes.

Enter Laplace, the prince of determinists, the man who wrote Mecanique Celeste, who, when on presenting the work to Napoleon was asked why in none of the five volumes had he mentioned the Author of the universe, replied, “Sire, I had no need of that hypothesis.” Laplace was the man who developed the theory of perturbations. the tech­nique of computing the future or past positions of any given planet with acceptable precision in spite of the inaccuracies inherent in analysing the Using these techniques Adams and Le Verrier independently analysed the perturbations of Uranus and predicted the position of hitherto undiscovered Neptune. As we now know, the discov­ery owed more to luck than anything else, for a mass of the perturbing planet had to be assumed, and Adams chose 45 terrestrial masses and Le Verrier 32. The actual mass of Neptune is a mere 17 times the weight of Earth, and thus although the discovery is still quoted as a demonstration of the power of celestial mechan­ics, it is in fact no more than an example of fortune favouring the system as a set of two body problems.

In spite of its limita­tions, the successes of celestial mechanics were used to buttress the dogma that science was on its way to explaining everything, that its powers of predic­tion were on the verge of being faultless. That is to say that the predictive models were on the verge of being more than just very good but perfect. Soon they would be capable of projection forward into the infinitely distant future or back to the beginning of time, and capable of perfect accu­racy. This vaunting self-confidence was to be destroyed at the end of the nineteenth century by the work of the French mathematician Henri Poincare, who proved that there is no exact general solution to the Three Body Problem, let alone the N Body Problem. That is, that the Kepler/Newton programme was in princi­ple unachievable.

This is not to say that the solar system is com­pletely chaotic, even in the long term, but rather that it is not completely deter‑ministic. Recent studies have demonstrated that over periods as long as a hundred million years, there is no sign of gross instability. Each planet orbits within a toroidal volume around the Sun, never going beyond the boundaries of that figure, yet in the long term never predictably positioned within it.

Such a situation holds for bodies whose orbital periods are not simple ratios of one another, such as 1:1, 1:2, 2:3, etc. With such ratios, the objects are said to be in resonance, and the situation is either completely stable or grossly unstable depending upon the resonant ratio and the relative positions of the objects. Although the orbital periods of the planets are not resonant, those of many asteroids are, and this is why from time to time one becomes so chaotic as to leave the belt and either escape from the solar system or move in towards the Sun on an Earth-crossing orbit.

Evidence of this process and the fact that it is still going on is seen when the average distance from the Sun of all known asteroids is plotted against the number of individuals in any zone. The histogram shows very small, but not zero, populations at resonant ratios with Jupiter of 3:1, 5:2 and 2:1. These zones of instability are called Kirkwood Gaps and they mark the origin of many of the bodies which bombard the inner planets. Two asteroids are known to inhabit the 3:1 gap, and 1989AC, which is due to pass within .011 A.U. of the Earth (four times the distance to the Moon) in 2004, was probably from here also.

A background of naive determinism, the dogma of scientific infallibility and the belief that mankind is the static pinnacle of evolution lies uneasily with the facts revealed by chaos theory. We are now forced to see that there is inherent uncertainty at the macro- as well as the microscopic scale.

Einstein believed that God does not play dice, but we now know that in this he was wrong. In fact, not only does God play dice, but also in this corner of the cosmos he plays bowls as well. However, the calculated power spectrum of impacts gives no great cause for alarm. Although Earth is under continuous bombardment by micro-meteorites, the strike rate drops rapidly with size. Nation-destroying colli­sions average only one every 300,000 years, and death-of-the-dinosaurs strikes as infrequently as every 60,000,000 years.

In the intermediate range, “city smackers” like the Tunguska object (which devastated over a thousand square kilometres of Siberian wilderness in 1908) arrive at the rate of one every 300 years. This may seem unpleasantly frequent, but given the area of the Earth and the area of devastation, the likelihood of any given individual’s name being on it is very small indeed.

A New Zealander is much more likely to be fatally surprised by falling masonry in an earthquake than by an interplanetary brickbat.